Week 5 · Similarity of polygons, the scale factor and the AA postulate for triangles · 3 lessons
2
Finding missing sides, the ratio of perimeters, enlargement or shrinking and the drawing scale
Mathematics Sec 1 Term 1, Unit 2, Lessons 1–2
GoalUse the scale factor to find sides and perimeters, and to read a drawing scale.
New words
enlargement · a similar figure with k > 1 (the first figure is bigger)drawing scale · the ratio of a length in the drawing to the real length, such as 1 : 2000perimeter · the total length around a polygon
Explanation
If two polygons are similar with scale factor k, every pair of corresponding sides has ratio k, and so the ratio of the perimeters is also k. Proof: if AB/EF = BC/FG = … = k, then by the property of proportion (sum of numerators over sum of denominators) (AB + BC + …)/(EF + FG + …) = k. So when you know the perimeters you also know k, and then any side. To enlarge a figure by a factor k multiply every length by k (angles do not change); to shrink it, multiply by a fraction between 0 and 1. A drawing scale such as 1 : 2000 is a scale factor: 1 cm on the drawing stands for 2000 cm = 20 m in real life. Convert everything to the same unit before you divide.
Shape
Scale factor k
Meaning
enlargement
k > 1
the first is bigger
shrinking
0 < k < 1
the first is smaller
congruent
k = 1
same size
Worked examples
△ABC ~ △DEF. DE = 6, EF = 8, FD = 10 (perimeter 24) and the perimeter of △ABC is 60. Then k (ABC to DEF) = 60/24 = 5/2. So AB = 6 × 5/2 = 15, BC = 8 × 5/2 = 20, CA = 10 × 5/2 = 25. Check: 15 + 20 + 25 = 60 ✓.
Shrink a 20 cm × 12 cm rectangle by the factor 0.4. The new sides are 20 × 0.4 = 8 cm and 12 × 0.4 = 4.8 cm. The perimeter goes from 2 × 32 = 64 cm to 25.6 cm, and 64 × 0.4 = 25.6 ✓.
A football pitch is 100 m by 60 m. Draw it at the scale 1 : 2000. Convert to centimetres: 100 m = 10 000 cm and 60 m = 6000 cm. Divide by 2000: the drawing is 5 cm by 3 cm.
Class activity
Scale it yourself. Measure your exercise book (length and width in cm). Find the sides of a similar rectangle with k = 2 and with k = 1/2, and the new perimeters.
Practice
1
A triangle has sides 4, 5 and 7 cm. Enlarge it with k = 3. Give the sides and both perimeters.
Sides 12, 15, 21 cm. Perimeters 16 cm and 48 cm (16 × 3).
2
Two similar polygons have perimeters 36 cm and 24 cm. A side of the first is 15 cm. Find the matching side of the second.
k = 36/24 = 3/2, so the side is 15 ÷ 3/2 = 10 cm.
3
A photo is 12 cm by 9 cm. It is enlarged keeping its shape so that the width becomes 36 cm. Find the new height.
k = 36/12 = 3 (from the new to the old), so the new height is 9 × 3 = 27 cm.
4
A school yard is 40 m long. How long is it on a drawing at the scale 1 : 500?
40 m = 4000 cm and 4000 ÷ 500 = 8 cm.
5
Two similar triangles have perimeters 48 cm and 30 cm. A side of the smaller is 10 cm. Find the matching side of the larger.
The factor larger : smaller = 48/30 = 8/5, so the side is 10 × 8/5 = 16 cm.
